How many times in a day, the two hands of a clock are in a straight line but not together

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1. Minute Spaces

The face or dial of clock is a circle whose circumference is divided into 60 equal parts, named minute spaces.

2. Hour hand and minute hand

A clock has two hands. The smaller hand is called the hour hand or short hand and the larger one is called minute hand or long hand.

3. In 60 minutes, minute hand gains 55 minute spaces over the hour hand.

(In 60 minutes, hour hand will move 5 minute spaces while the minute hand will move 60 minute spaces. In effect the space gain of minute hand with respect to hour hand will be 60 - 5 = 55 minutes.)

4. Both the hands of a clock coincide once in every hour.

5. The hands of a clock are in the same straight line when they are coincident or opposite to each other.

6. When the two hands of a clock are at right angles, they are 15 minute spaces apart.

7. When the hands of a clock are in opposite directions, they are 30 minute spaces apart.

8. Angle traced by hour hand in 12 hrs = 360°

9. Angle traced by minute hand in 60 min. = 360°.

10. If a watch or a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast.

11. If a watch or a clock indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.

12. The hands of a clock will be in straight line but opposite in direction, 22 times in a day.

13. The hands of a clock coincide 22 times in a day.

14. The hands of a clock are straight 44 times in a day.

15. The hands of a clock are at right angles 44 times in a day.

16. The two hands of a clock will be together between $H$ and $(H+1)$ o' clock at
$\left(\dfrac{60H}{11}\right)$ minutes past $H$ o' clock.

17. The two hands of a clock will be in the same straight line but not together between $H$ and $(H + 1)$ o' clock at

$\begin{cases} (5H - 30)\dfrac{12}{11} \\ \text{ minutes past }H \text{, when } H\gt 6\\~\\ (5H + 30)\dfrac{12}{11} \\ \text{ minutes past }H \text{, when } H\lt 6 \end{cases}$

18. Angle between hands of a clock When the minute hand is behind the hour hand, the angle between the two hands at $M$ minutes past $H$ 'o clock $=30\left(H - \dfrac{M}{5}\right) + \dfrac{M}{2}$ degree When the minute hand is ahead of the hour hand, the angle between the two hands at $M$ minutes past $H$ 'o clock

$=30\left(\dfrac{M}{5} - H\right) - \dfrac{M}{2}\text{ degree}$

19. The two hands of the clock will be at right angles between $H$ and $(H+1)$ o' clock at
$(5H \pm 15)\dfrac{12}{11}$ minutes past $H$ 'o clock

20. If the minute hand of a clock overtakes the hour hand at intervals of $M$ minutes of correct time, the clock gains or loses in a day by
$\left(\dfrac{720}{11} - M \right)\left(\dfrac{60 \times 24 }{M}\right)$ minutes

21. Between $H$ and $(H+1)$ o' clock, the two hands of a clock are $M$ minutes apart at
$(5H \pm M)\dfrac{12}{11}$ minutes past $H$ 'o clock

How to find whether the clock loses or gain? For example:

The minute hand of a clock overtakes the hour band at interval of 60 mins. How much a day does the clock gain or lose?

Let me explain with two examples.

example 1: The minute hand of a clock overtakes the hour band at interval of $60$ mins. How much a day does the clock gain or lose?

Since the sign is $\text{-ve}$, clock loses $\dfrac{1440}{121}\text{ min}$ a day.

Angle between Min hand and Hours hand
Ɵ = (11/2)M-30H

A clock shows the time as 6 a.m. If the minute hand gains 2 minutes every hour, how manyminutes will the clock gain by 9 p.m.?

Given It gains  2 minutes a hourThen,Between 6am and  9 am = 3 hrs I.e 3*2=6Between 9 am and 9 pm it is 12 hrs then 12*2=24

6+24=30

total 15 hours
so 15 * 2 = 30 min

Assume that the clocks shows the correct time at 6 am.2 minutes are gained in every hr.

from 6 am to 9 pm,  15 hours are there and 15*2 = 30 minutes are gained.

In a clock min. Hand cross hr. Hand in 65min. Find out the clock fast and slow by how many min?

Use any of the following formula

formula 1

if you explain with example then it will be better

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